(i) Since \({\cal C}_o\rightarrow {\cal C}\)  preserves all ordinary (co)limits we can use [1]} to break down the extension of \(JH\)  along \(J\)  into first extending \(J_oH\)  along \(J_o\)  to \(\tilde{H}\)  and then extending \(\tilde{H}\)  to \(H^{\prime }\) .
(ii) If a functor \({\cal C}_o\rightarrow {\cal C}_o\)  is locally monotone then this functor is its own extension (both left and right) to \({\cal C}\rightarrow {\cal C}\) . This means that for locally monotone functors \(\tilde{H}\)  the upper square is trivial.
(iii) To compute \(\tilde{H}\)  we can use (REF ) or (REF ), and (REF ). — While this is sometimes a good approach, the downside is that \(\tilde{H}\)  is typically not locally monotone (so we cannot use (i)) and the inclusion \({\cal C}_o\rightarrow {\cal C}\)  is not fully faithful (so we loose the good properties of Kan extensions along fully faithful functors). To summarise, to compute Kan extensions along \(\mathsf {D}\)  we use (REF ) and for \(\mathsf {W}\)  we will develop a similar presentation in Theorem REF .
